In a large skillet or shallow heavy pot, heat oil over medium-high. Add onion and sausages and cook until sausages are deep golden brown, 6 minutes, turning sausages and stirring onion halfway through. Add sage and cider and season with salt and pepper. Bring to a rapid simmer and cook until cider is syrupy and coats the back of a spoon, about 30 minutes.
Meanwhile, in a medium saucepan combine squash, potato, and apple. Cover by 2 inches water and bring to a rapid simmer over medium-high. Cook until squash and potato are tender when pierced with a knife, 15 minutes.
Drain vegetables and apple and transfer to a food processor. Process until smooth. Add yogurt, season with salt and pepper, and pulse to combine. Divide butternut mash among four plates and serve with sausages and onion.